23.5.84 problem 84

Internal problem ID [6693]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 84
Date solved : Friday, October 03, 2025 at 02:09:45 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} a \,x^{2} y+6 y^{\prime }+6 x y^{\prime \prime }+x^{2} y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 57

ode:=a*x^2*y(x)+6*diff(y(x),x)+6*x*diff(diff(y(x),x),x)+x^2*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \,{\mathrm e}^{\frac {\left (-a \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-a \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2}}+c_3 \,{\mathrm e}^{\left (-a \right )^{{1}/{3}} x}}{x^{2}} \]

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 58

ode=a*x^2*y[x] + 6*D[y[x],x] + 6*x*D[y[x],{x,2}] + x^2*D[y[x],{x,3}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 e^{-\sqrt [3]{a} x}+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x}+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x}}{x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*x**2*y(x) + x**2*Derivative(y(x), (x, 3)) + 6*x*Derivative(y(x), (x, 2)) + 6*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(-a*x*y(x) - x*Derivative(y(x), (x, 3)) - 6*Derivative(y(x),